Number of integral values of n for which limit
The number of non negative integral values of n for which lim x=0 (cosx-1)*(cosx-e^x)/x^n.
I tried by series expansion but not getting anything useful. How to proceed?
Plz help
You can taylor exapand around $0$ and get: $$ \lim_{x \to 0}\frac{(1-x^2/2-1)(1-x^2/2-1-x-x^2/2)}{x^n}\\ = \lim_{x \to 0} \frac{\frac{x^2}{2}(x+x^2)}{x^n}\\ = \lim_{x \to 0}\frac{1}{2}(x^{4-n}+x^{3-n}) $$ from this you see that if $n>3$ it goes to infinity, in $n =3$ it goes to $1/2$ if $n<3$ it goes to $0$.